Animating separate objects with the same armature
The body and the armour are separate objects and you only have the armature moving the body, you also need to have the armature linked to the armour as well.

One way is to parent using automatic weights or manual weight painting, the same as you would for the body.

As your armour is most likely to be a single rigid piece you might want to parent the armour to a single bone so that it stays as one solid piece. To do this select the armour, shift select the armature and press CtrlP then select Bone. The bone used will be the selected bone in pose mode, you can change to pose mode and select the right bone before parenting.



EDIT: As you want to export to unity I would join the two meshes and use weight painting instead of parenting.

Select the armour and then the body and press CtrlJ. This will join the two objects into one mesh. Selecting the body last means the armour will be added to the body, keeping the armature setup as it is.

Go into edit mode and select one of the armour vertices. Press CtrlL to select the rest of the connected vertices. In the vertex groups find Chest and click assign with the weight set to 1.0. This will make the entire armour piece move as one piece as the chest bone is moved.